Student perspectives on equation: The transition from school to university

Abstract

It has been recognised that many student perspectives on equations and their use of the equals sign have not mirrored those that mathematicians would like to see in tertiary students. This paper tracks transition of understanding of the equals sign by comparing secondary school students' thinking with that of first year university students. We analyse the understanding displayed in terms of properties of the constituent parts of equations, identifying a number of incomplete or pseudoconceptions that are sometimes influenced by representational aspects of the properties, and other times by apparent over-generalisation of a property. A start is made on constructing a framework for understanding of the mathematical equation object that could assist in the transition from school to tertiary mathematics study.

Key Mathematical Concepts in the Transition from Secondary School to University

“…The study showed that constructs of number, symbolic literals, operators, the '=' symbol itself, and the formal equivalence relation, as well as the principles of arithmetic, all contribute to building a deep understanding of equation. This agrees with the observations of Godfrey and Thomas (2008), who, using the TWM framework, provided evidence that many students have a surface structure view of equation and fail to integrate the properties of the object with that surface structure.…”

Mathematical Prerequisites for STEM Programs: What do University Instructors Expect from New STEM Undergraduates?

“…In more detail, new STEM undergraduates seem to employ surface learning and understanding, to be inflexible in switching between the embodied, symbolic and formal worlds, to focus on an exclusive process view and to apply only intuitive thinking (e.g., Godfrey and Thomas 2008 for abstract algebra; Thomas and Stewart 2011 for linear algebra; Carlson et al 2015 for calculus and analysis). For instance, students were reported to have a limited concept of equality that is not based on a formal understanding of equivalence relation and its properties (Godfrey and Thomas 2008) and to view functions as a picture of an event and recipe to get an answer instead of two quantities changing together and a mapping of input values of the function's domain to output values in the function's range (summarized by Carlson et al 2015). Students were also found to have an unstable conceptualization of slope preventing covariational reasoning (Nagle et al 2013;Thompson 1994).…”

“…For example, with regard to mathematical content aspects, students are expected to know about the properties of equivalence relation and implication. Such knowledge is viewed as a foundation to interact fully with the equation sign (Godfrey and Thomas 2008). In line with considerations of Carlson et al (2015), the university instructors expect that students have a concept image of mathematical functions that corresponds with the concept definition and thus enables covariational reasoning.…”

Versatile thinking and the learning of statistical concepts

“…In the same way, a table of values, a bar graph or a box plot all elicit particular examples of concepts in the sense that they are counterparts, in much the same way as a symbol such as P f Áx P f evokes the concept of the mean. Of course, this use of various symbols as counterparts is not automatic, but requires an individual to interpret them by 'overlaying' an appropriate schema (Godfrey & Thomas, 2008). This can be seen in the case of the difference between a drawn representation of a circle and the mathematical figure, or object, of the circle of which it is a counterpart (Booth & Thomas, 2000).…”